The New York system was never really "private." The IRT and BMT were designed by the city, mostly built by the city, and were leased to the private operators. The IND was always municipally owned and operated. The city bought all of the systems in 1940, about 35 years after the first one was built.

That is not to say you can't have private systems that work (Tokyo, Hong Kong). New York just isn't a particularly good example.